A065656 Composite numbers k such that sigma(k)*phi(k) + 2*(k+1) is a square.

1169, 7777, 41111, 46097, 668167, 846817, 2107519, 3612769, 17424241, 30666527, 37526993, 56323393, 214746055, 383523857, 512376769, 1021934641, 1228492849, 1303949599, 4056001351, 7425397169, 17073544447, 17859428369, 18452226887, 46874737969, 51411954391

Offset: 1

Author

Labos Elemer, Nov 12 2001

Keywords

• nonn

Comments

a(n) and square root of phi(a(n))*sigma(a(n)) + 2*a(n) + 2 are close to each other: e.g., a(7) = 2107519 and this square root is 2107458.

Since (p+1)*(p-1) + 2*(p+1) = p*p + 2*p + 1 = (p+1)^2 is a square, all primes are solutions.

73362272287 and 181264312447 are also terms. - Donovan Johnson, Jul 13 2012

Examples

			k = 7777: sigma(7777) = 9792, phi(7777) = 6000 and 9792*6000 + 2*7778 = 587675556 = 7666^2.
		

Crossrefs

Cf. A062354, A000203, A000010, A065655.

Programs

PARI

isok(k) = { !isprime(k) && issquare(sigma(k)*eulerphi(k) + 2*(k + 1)) } \\ Harry J. Smith, Oct 26 2009

Extensions

a(9)-a(15) from Harry J. Smith, Oct 26 2009

a(16)-a(20) from Donovan Johnson, May 24 2011

a(21)-a(25) from Donovan Johnson, Jul 13 2012